Clean button mushrooms and slice. Mince the shallot. Braise both in butter for approximately 5 minutes. Deglaze with white wine, cook and reduce liquid slightly.
Butter a gratin dish. Mix half of the mushrooms with herbs and distribute on the bottom of the dish.
Add salt and pepper to the filets and place on top the mushrooms. Cover with the remaining mushrooms and bake in the 350°F oven for approximately 20 minutes.
Stir egg yolk into the cream. Remove the juice that formed while baking the fish and add it to the egg-cream mixture. Season with pepper and salt and pour over the fish filets. Sprinkle cheese over the dish and add a few flakes of butter.
Bake another 10 minutes at 350°F heat. Don’t let a brown crust form, if necessary cover with aluminum foil.
Serve with freshly peeled boiled potatoes, grainy rice or fine noodles.